{
  "abstract": "Preference optimisation improves helpfulness, but a single aggregate safety penalty lets frequent harm categories dominate the objective while rare ones bear the residual risk. This paper models post-training as a vector-constrained programme: the pairwise preference loss is minimised under one risk tolerance per harm category. The solver is a projected primal–dual gradient recursion that prices each category by its own residual violation. For a nonconvex objective we bound three horizon-averaged residuals—primal stationarity, dual regret and the average constraint value—for the unbiased, unclipped recursion and a uniformly drawn iterate; the implemented recursion is compared empirically with a no-EMA, no-clipping form of itself. Certification stays separate from optimisation: on an untouched calibration split, a multiplicity-adjusted fixed-sequence test gives release thresholds valid for all categories at once under correct category assignment, with an additive bound for learned routers. It certifies the scoring and filtering of corpus-distributed candidate responses, not text the trained policy generates. In one exploratory setting—one Qwen2.5-0.5B-Instruct backbone, PKU-SafeRLHF, three seeds—the recursion meets the per-category tolerance that direct preference optimisation and scalar-constrained schemes violate, cutting the worst-category miss rate at the training threshold from 0.176 to 0.057 at unchanged preference accuracy. The certified operating point barely moves: most of the constraint’s effect is a score translation shared by the harmful and benign responses of a prompt, which calibration provably absorbs. Whether a training constraint holds and whether the deployed gate is safer are different questions.",
  "abstract_shown": true,
  "affiliations": [
    "School of Information Science and Engineering, Dalian Polytechnic University, Ganjingzi District, Dalian 116034, China",
    "General Graduate School, Yonsei University, Seoul 03722, Republic of Korea",
    "School of Technology and Management, Kyung Hee University, Seoul 02447, Republic of Korea"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Haochen Jiang",
      "orcid": "0009-0005-4322-9512"
    },
    {
      "affs": [
        1,
        2
      ],
      "name": "Yulin Tu",
      "orcid": "0000-0002-3165-6278"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dde5pv656d3mt.bib",
  "cited_by": 0,
  "doi": "10.3390/math14193638",
  "fulltext_url": null,
  "id": "dde5pv656d3mt",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/19/3638",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "1702|Adversarial Robustness in Machine Learning"
    ]
  ],
  "openalex": "W7220922959",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/dde5pv656d3mt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14193638",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/19/3638"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Projected Primal–Dual Gradient Preference Optimization Under Category-Wise Split-Conformal Safety Constraints for Large Language Model Post-Training",
  "title_text": "Projected Primal–Dual Gradient Preference Optimization Under Category-Wise Split-Conformal Safety Constraints for Large Language Model Post-Training",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/dde5pv656d3mt",
  "venue": "Mathematics",
  "versions": []
}